TY - GEN
T1 - Circular orders, ultra-homogeneous order structures, and their automorphism groups
AU - Glasner, Eli
AU - Megrelishvili, Michael
N1 - Publisher Copyright: © 2021 by the American Mathematical Society.
PY - 2021
Y1 - 2021
N2 - We study topological groups G for which either the universal minimal G-system M(G) or the universal irreducible affine G-system IA(G) is tame. We call such groups “intrinsically tame” and “convexly intrinsically tame”, respectively. These notions, which were introduced in [Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Springer, Cham, 2018, pp. 351-392], are generalized versions of extreme amenability and amenability, respectively. When M(G), as a G-system, admits a circular order we say that G is intrinsically circularly ordered. This implies that G is intrinsically tame. We show that given a circularly ordered set X◦, any subgroup G ≤ Aut (X◦) whose action on X◦ is ultrahomogeneous, when equipped with the topology τp of pointwise convergence, is intrinsically circularly ordered. This result is a “circular” analog of Pestov’s result about the extreme amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. We also describe, for such groups G, the dynamics of the system M(G), show that it is extremely proximal (whence M(G) coincides with the universal strongly proximal G-system), and deduce that the group G must contain a non-abelian free group. In the case where X is countable, the corresponding Polish group of circular automorphisms G = Aut (Xo) admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that M(G) = Split(T;Q◦), a circularly ordered compact metric space (in fact, a Cantor set) obtained by splitting the rational points on the circle T. We show also that G = Aut (Q◦) is Roelcke precompact, satisfies Kazhdan’s property T (using results of Evans-Tsankov), and has the automatic continuity property (using results of Rosendal-Solecki).
AB - We study topological groups G for which either the universal minimal G-system M(G) or the universal irreducible affine G-system IA(G) is tame. We call such groups “intrinsically tame” and “convexly intrinsically tame”, respectively. These notions, which were introduced in [Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics, Springer, Cham, 2018, pp. 351-392], are generalized versions of extreme amenability and amenability, respectively. When M(G), as a G-system, admits a circular order we say that G is intrinsically circularly ordered. This implies that G is intrinsically tame. We show that given a circularly ordered set X◦, any subgroup G ≤ Aut (X◦) whose action on X◦ is ultrahomogeneous, when equipped with the topology τp of pointwise convergence, is intrinsically circularly ordered. This result is a “circular” analog of Pestov’s result about the extreme amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations. We also describe, for such groups G, the dynamics of the system M(G), show that it is extremely proximal (whence M(G) coincides with the universal strongly proximal G-system), and deduce that the group G must contain a non-abelian free group. In the case where X is countable, the corresponding Polish group of circular automorphisms G = Aut (Xo) admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that M(G) = Split(T;Q◦), a circularly ordered compact metric space (in fact, a Cantor set) obtained by splitting the rational points on the circle T. We show also that G = Aut (Q◦) is Roelcke precompact, satisfies Kazhdan’s property T (using results of Evans-Tsankov), and has the automatic continuity property (using results of Rosendal-Solecki).
KW - Amenability
KW - Automatic continuity
KW - Circular order
KW - Extremely amenable
KW - Fraïssé class
KW - Intrinsically tame
KW - Kazhdan’s property T
KW - Roelcke precompact
KW - Thompson’s circular group
KW - Ultrahomogeneous
UR - http://www.scopus.com/inward/record.url?scp=85114844851&partnerID=8YFLogxK
U2 - https://doi.org/10.1090/conm/772/15486
DO - https://doi.org/10.1090/conm/772/15486
M3 - منشور من مؤتمر
SN - 9781470456641
T3 - Contemporary Mathematics
SP - 133
EP - 154
BT - Topology, Geometry, and Dynamics
A2 - Vershik, Anatoly M.
A2 - Buchstaber, Victor M.
A2 - Malyutin, Andrey V.
PB - American Mathematical Society
T2 - International Conference on Topology, Geometry, and Dynamics, 2019
Y2 - 19 August 2019 through 23 August 2019
ER -